CHAP, xx] PBOPERTIES OF THE DIGITS OF NUMBERS. 463
D. Biddle61 applied congruences to find nuihbers like 15 and 93 whose product 1395 has the same digits as the factors.
P. Cattaneo62 considered numbers Q (and C) whose square (cube) ends with the same digits as the number itself. No Q> 1 ends with 1. No two <2's with the same number of digits end with 5 or with 6. All Q's < 1014 are found. A single C of n digits ends with 4 or 6. Any Q is a C. Any Q—l is a C. If TV" is a Q with n digits and if 2N— I has n digits, it is a C.
M. Thi<§,62° using all nine digits >0, found numbers of 2, 3 or 4 digits with properties like 12-483 = 5796.
Pairs626 of cubes 33, 66 and 3753, 3873 whose sums of digits are squares, 32 and 62.
T. C. Lewis63 discussed changes in the digits of a number to base r not affecting its divisibility by p.
Numbers64 B and jBn having the same sum of digits.
Pairs65 of primes like 23-89 = 29-83.
Cases66 like 7-9403 = 65821 and 3-1458 = 6-0729, where the digits 0, 1,. . ., 9 occur without repetition.
JVpn+1 ending67 with the same digits as N.
Numbers68 like 512=(5+l+2)3, 47045881000000 = (47 +4 +58 +81)8.
All69 numbers like 2-5-27 = 1-18-15, 2+5+27 = 1+15+18.
Number70 divisible by the same number reversed.
Number71 an exact power of the sum of its digits; two numbers each an exact power of the sum of the digits of the other.
Solve72 KN+P=Nr, N' derived from N by reversing the digits.
Symmetrical numbers (ibid., p. 195).
F. Stasi73 proved that, if a, 6 are given integers and a has m digits, we can find a multiple of 6 of the form
Taking b prime to a and to 10, we see that 6 divides 10"*+ ... +1. The case m = 1 gives the result of Plateau.15
Cunningham73" and others wrote N\ for the sum of 'N and its digits to base rf N% for the sum of N\ and its digits, etc., and found when Nm is divisible by r— 1.
"Math. Quest. Educ. Times, (2), 19, 1911, 60-2. Cf. (2), 17, 1900, 44.
«Periodico di Mat., 26, 1911, 203-7.
•"Nouv. Ann. Math., (4), 11, 1911, 46.
"6Sphinx-Oedipe, 6, 1911, 62.
"Messenger Math., 41, 1911-12, 185-192.
wL'interme'diaire des math., 18, 1911, 90-91; 19, 1912, 267-8.
«TOd., 1911, 121, 239. "Ibid., 19, 1912, 26-7, 187.
<Vbid., 50-1, 274-9.
"Ibid., 77-8, 97.
*»Ibid., 125, 211.
™Ibid., 128.
nlbid., 137-9, 202; 20, 1913, 80-81.
"Ibid., 221.